Problem 19 - Calculus Early Transcendentals

Question

The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of \(f\) and the \(x\) -axis on the interval using a left, right, and midpoint Riemann sum with \(n=4\) c. Use the sketch in part (a) to show which intervals of \([a, b]\) make positive and negative contributions to the net area. $$f(x)=\tan ^{-1}(3 x-1) \text { on }[0,1]$$

Step-by-Step Solution

Answer

Question: Sketch the function f(x) = tan^(-1)(3x-1) on the interval [0,1] and determine the intervals that contribute positively and negatively to the net area. Answer: The function f(x) = tan^(-1)(3x-1) contributes negatively to the net area in the interval [0, a] and positively to the net area in the interval [a, 1].

Step 1: Sketch the Function

First, we will sketch f(x) = \(\tan^{-1}(3x-1)\) on the interval [0,1] by plotting a few key points and connecting them. You can use a graphing calculator or any online graphing tool to plot the function.

Step 2: Approximate the Net Area using Riemann Sums

We will now approximate the net area bounded by the graph of \(f(x)\) and the \(x\)-axis on the interval [0,1] using left, right, and midpoint Riemann sums using 4 sub-intervals. 1. Left Riemann Sum: Choose the height of each rectangle as the value of the function at the left endpoint of the sub-interval. Calculate the width of the interval by finding \(\Delta x = \frac{1-0}{4} = 0.25\). Left Riemann Sum: $$LRS = \Delta x (f(0) + f(0.25) + f(0.5) + f(0.75))$$ 2. Right Riemann Sum: Choose the height of each rectangle as the value of the function at the right endpoint of the sub-interval. Right Riemann Sum: $$RRS = \Delta x (f(0.25) + f(0.5) + f(0.75) + f(1))$$ 3. Midpoint Riemann Sum: Choose the height of each rectangle as the value of the function at the midpoint of the sub-interval. Midpoint Riemann Sum: $$MRS = \Delta x (f(0.125) + f(0.375) + f(0.625) + f(0.875))$$ Now, calculate the function values and evaluate each Riemann sum.

Step 3: Determine Positive and Negative Contributions to Net Area

To determine the intervals that contribute positively and negatively to the net area, examine where the function \(f(x)\) is positive or negative between 0 and 1. 1. Analyzing the function and the graph, we can see that when x approaches 0, the function value approaches \(\tan^{-1}(3(0)-1)=-\tan^{-1}(1)<0\). 2. When x approaches 1, the function value approaches \(\tan^{-1}(3(1)-1)=\tan^{-1}(1)>0\). 3. The function has a root (in which the function value is zero) between 0 and 1 where \(\tan^{-1}(3x-1)=0\), so the transition from negative to positive occurs. From these observations, we can conclude that \(f(x)\) contributes negatively to the net area in the interval \([0, a]\) and positively to the net area in the interval \([a, 1]\).

Key Concepts

Function Sketching

Sketching a function is a crucial step in understanding its behavior over a given interval. For the function \(f(x) = \tan^{-1}(3x-1)\) on the interval \([0, 1]\), we begin by plotting key points to form a comprehensive graph.

Here's how to approach it:

• Start by calculating the function values at significant points like the interval's endpoints and midpoints.
• For example, evaluate the function at \(x = 0, 0.25, 0.5, 0.75,\) and \(1.0\).
• When \(x = 0\), the function evaluates to \(\tan^{-1}(-1)\), which is negative.
• When \(x = 1\), it evaluates to \(\tan^{-1}(2)\), which is positive.

Plot these points and use a smooth curve to connect them. This graph will help visualize where the function is positive and negative within the interval.

Net Area

To approximate the net area under the curve of \(f(x) = \tan^{-1}(3x-1)\), we employ the concept of Riemann sums. The net area in this context refers to the total area bounded by the function and the \(x\)-axis, taking into account sign changes over the interval.

We consider three types of Riemann sums:

• Left Riemann Sum: It uses the function's value at the left endpoint for each sub-interval. Calculate the height at \(x = 0, 0.25, 0.5,\) and \(0.75\).
• Right Riemann Sum: It uses the right endpoint of each sub-interval. Evaluate the height at \(x = 0.25, 0.5, 0.75,\) and \(1.0\).
• Midpoint Riemann Sum: It uses the function value at the midpoint of each sub-interval, such as \(x = 0.125, 0.375, 0.625,\) and \(0.875\).

By computing each, we can estimate how much of the area is above or below the \(x\)-axis, contributing to a positive or negative net area.

Arctangent Function

The arctangent function, denoted \(\tan^{-1}(x)\), is the inverse of the tangent function and has special properties. It outputs an angle whose tangent equals the given input \(x\). Within the context of our function \(f(x) = \tan^{-1}(3x-1)\), this means we are examining how the expression \(3x-1\) translates into angles.

Understanding \(\tan^{-1}\) helps us:

• Predict how the function behaves as \(x\) changes over \([0, 1]\).
• Anticipate the transition points where the function changes from negative to positive, since the arctangent of zero is zero.
• Identify critical points of transformation on the graph.

The arctangent function continues to increase but asymptotically approaches \(\pi/2\) and \(-\pi/2\) as its input grows large positive or negative, respectively. This behavior helps predict and interpret graph transformations and contributes to understanding the overall area computations.